The need to find the inverse of a matrix in matrix algebra is immense. This brings us to one of the easiest ways to find the inverse of a matrix – using elementary row operations.

## What are elementary operations of a matrix?

- Interchange of any two rows or two columns. Symbolically written as, and
- The multiplication of the elements of any row or column by a non-zero number. Symbolically shown as, and
- The addition to the elements of any row or column, the corresponding elements of any other row or column multiplied by any non zero number. Represented as, and

## Invertible Matrices

A square matrix **A** is said to invertible if its determinant has a non-zero value, i.e., **A** is non-singular.

For a square matrix **A** of order n, its inverse is denoted by and has an order n. The product of a matrix and its inverse results to the Identity matrix of order n. We can observe that, *if B is the inverse of A, then A is the inverse of B.*

The notion of inverse can be used to calculate an unknown matrix within multiplication.

*Example:*

To know the values of **X**, we need to multiply both sides of the equation with the inverse of A. Here, the order of X, A, B is same.

If A and B are known, then X can be calculated using above steps.

## Elementary row operations – Inverse of matrix

Before diving into calculating the inverse of a matrix, let us know more about what operations should be applied to the product AB, if elementary operations (transformations) are performed on X, so that the equation X = AB holds. Row operations carried out on X should also be performed on the first matrix, A, of the product AB. Similarly, if column operations were carried out on X then they should also be carried on the second matrix, B, of the product B.

**Procedure**

1. We write , and then apply a sequence of row operations on A till we get, .

Example:

Using elementary operations, calculate the inverse of the matrix

Solution:

We start by writing .

We have been able to write as where B is the inverse of matrix A.

Hence, the inverse is

We can validate the answer by multiplying it with A, as .

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